The diagram shows the graphs of the functions $f(x)$ and $g(x)$ - HSC - SSCE Mathematics Extension 1 - Question 4 - 2023 - Paper 1

To find the area between the curves $y = f(x)$ and $y = g(x)$ from $x = a$ to $x = c$, we can evaluate the definite integrals given:

1. We know that: $\int_{a}^{c} f(x) \, dx = 10$

2. The area under $g(x)$ from $a$ to $c$ can be found by combining the areas from $a$ to $b$ and $b$ to $c$: $\int_{a}^{c} g(x) \, dx = \int_{a}^{b} g(x) \, dx + \int_{b}^{c} g(x) \, dx = -2 + 3 = 1.$

3. The area between the curves is thus:

   $\text{Area} = \int_{a}^{c} f(x) \, dx - \int_{a}^{c} g(x) \, dx$

   Substituting the known values:

   $\text{Area} = 10 - 1 = 9.$

Therefore, the area between the curves is 9.